1) Show us some of these photos. It's almost impossible to figure out what's wrong if you don't.
2) The Tamron 150-600 doesn't sound like it's going to help you much. In fact it will make things worse. It's even slower than your current lens. If you wanted to help with a new lens, you'd probably want something more like a 300mm f/4 (for roughly the same price), which would let you take the exact same shots with ISO 800 instead, and probably much Much MUCH sharper, since it's a prime instead of a crazy 17x zoom... Also, either of those lenses probably has better autofocus than your superzoom (more accurate)
3) It sounds like manual mode is probably not for you if you are confused by it to the point of not being able to get a proper exposure. I doubt this is the source of your main problems, but just as a side point, it's probably ONE factor holding you back unnecessarily. Manual mode can be good for some people, but if you don't know what the right decisions are yet / are confused (hence this thread), then you're probably not better off by being in control of everything just yet.
4) Are you sure 1/2000 is actually necessary to freeze a bird? They are generally a lot easier to freeze than something like a running mammal, since their body parts don't move as much in relation to one another. And are they even in flight? I'm not sure if you said. If they're perched, then you certainly don't need 1/2000
A faster lens and the possibility of not needing to go quite so fast might buy you 2 stops, + better optics, which might be enough to eliminate your noise and your soft images, respectively.
But again, post images first. It might be something much simpler than that.